Acylindrical actions for two-dimensional Artin groups of hyperbolic type

Abstract

For a two-dimensional Artin group A whose associated Coxeter group is hyperbolic, we prove that the action of A on the hyperbolic space obtained by coning off certain subcomplexes of its modified Deligne complex is acylindrical and universal. As a consequence, we obtain the Tits alternative for A, and we classify its subgroups that virtually split as a direct product. We also extend the classification of maximal virtually abelian subgroups to arbitrary two-dimensional Artin groups. A key ingredient in our approach is a simple criterion to show the acylindricity of an action on a two-dimensional CAT(−1) complex.

Parabolic subgroups of large-type Artin groups

“…Indeed, several complexes have been associated to Artin groups using the combinatorics of parabolic subgroups. For instance, Deligne complexes and their variants are built out of (cosets of) standard parabolic subgroups of spherical type (Charney & Davis, 1995), and have been used to study various aspects of Artin groups: K(π, 1)-conjecture (Charney & Davis, 1995, Paris, 2014, acylindrical hyperbolicity (Charney & Morris-Wright, 2019, Martin & Przytycki, 2019, Vaskou, 2020, Tits alternative (Martin & Przytycki, 2020), etc. More recently, using the connections between braid groups and mapping class groups, the irreducible parabolic subgroups have been used to define a possible analogue of the complex of curves for Artin groups of spherical type (Cumplido et al, 2019, Morris-Wright, 2021.…”

Tits Alternative for groups acting properly on $2$-dimensional recurrent complexes (with an appendix written jointly with Jon McCammond)

“…In Appendix A written jointly with Jon McCammond we extend Theorem B to a class of 2-dimensional Artin groups containing all large-type Artin groups. In the case where G = A Γ we will extend Theorem B to all 2-dimensional A Γ with W Γ hyperbolic in a forthcoming article [MP19]. We will give there an account on the current state of affairs concerning the Tits Alternative for other classes of Artin groups.…”

Acylindrical hyperbolicity of Artin groups associated with graphs that are not cones

“…Many affirmative partial answers for Problem 1.2 are known. Indeed the following irreducible Artin groups of infinite type are known to be acylindrically hyperbolic: Right-angled Artin groups ( [9], [22]); Two-dimensional Artin groups such that the associated Coxeter groups are hyperbolic ( [23]); Artin groups of XXL-type ( [17]); Artin groups of type FC such that the defining graphs have diameter greater than 2 ( [12]); Artin groups that are known to be CAT(0) groups by a result of Brady and McCammond ([4], [21]); Euclidean Artin groups ( [7]). Actually, except for Euclidean Artin groups, all of these Artin groups are regarded as special cases of the following irreducible Artin groups of infinite type, which are known to be acylindrically hyperbolic: Artin groups associated to graphs that are not joins ( [11]); Two-dimensional Artin groups, that is, Artin groups such that every triangle with three vertices v 1 , v 2 , v 3 of the defining graphs satisfies…”